lcls-ii-tn-20-05 · 2020. 7. 28. · lcls-ii technical note july 28, 2020 lclsii-tn-20-05 4 given...

L C L S - I I T E C H N I C A L N O T E

July 28, 2020 LCLSII-TN-20-05 1

Non-linear compression with octupoles for current

profile shaping

LCLS-II-TN-20-05

7/28/20 N. Sudar, K. Bane, Y. Ding, Y. Nosochkov and Z. Zhang

SLAC, Menlo Park, CA 94025, USA


July 28, 2020 LCLSII-TN-20-05 2

Abstract: In this technical note we present a method for controlling the higher order compression in the LCLS-II bunch

compressors. An octupole magnet embedded in a chicane provides additional U5666, allowing for

suppression of current horns and additional shaping of the electron beam current profile. By adjusting the

octupole field strengths and electron beam betatron phase advance between octupoles embedded in

subsequent bunch compressors, projected emittance growth from the first octupole is corrected.

1 Introduction In order to achieve the electron beam peak current required by many applications, high brightness linear

accelerators employ multistage bunch compression. However, the peak current is limited by detrimental

higher order compression from RF curvature, longitudinal space charge (LSC), coherent synchrotron

radiation (CSR), and longitudinal wakefields. Although second order compression is typically compensated

with a harmonic cavity [1], higher order compression often remains unchecked. This can lead to the

production of horns in the current profile as the head and/or tail are over-compressed. These current horns

can produce significant LSC modulation, CSR and longitudinal wakefields downstream, causing energy

spread and projected emittance growth. This is particularly problematic in the LCLS-II linac where the

electron beam must be transported from the Linac exit through a 2 km bypass line.

We present here a scheme for controlling the higher order compression, effectively suppressing current horns

while providing additional knobs for shaping the current profile. An octupole magnet placed in a bunch

compressor at a point of high transverse dispersion will provide a transverse kick correlated with z³. After

transport through the remainder of the bunch compressor, this kick is converted to third order compression,

adjusting the U5666 of the bunch compressor. This scheme was investigated in [2-4], showing effective

suppression of current horns. However, significant octupole strength will lead to growth of the projected

emittance as the z dependent transverse kick will remain after compression. By installing a second octupole

in a downstream bunch compressor, this kick can be compensated, while providing additional U5666, provided

that the betatron phase advance and second octupole strength are chosen such that the kick is equal and

opposite, Figure 1.

This scheme has been studied for the LCLS-II superconducting linac. Elegant [5] simulations of several

Figure 1: Basic layout of the proposed scheme


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potential configurations are shown with an incoming electron beam from injector simulations using a cathode

laser with flattop and Gaussian temporal profile, section 2 and 3. A more detailed discussion of the

longitudinal shaping scheme and emittance correction is presented in section 4 and 5 respectively. A

discussion of minimization of particle losses after the first octupole is given in section 6.

2 Cathode laser with flattop temporal profile For the case where the electron beam is produced by a cathode laser with a flattop temporal profile, current

horns will naturally occur due to the third order chirp acquired in the injector and downstream collective

effects. In order to study the proposed scheme we consider an initial distribution generated by Impact

simulations of the LCLS-II injector [6-8]. Initial beam parameters are given in Table 1. The longitudinal

phase space at the entrance of the first linac section is shown in Figure 2 for 100pC and 50 pC beams. Note

that, according to Elegant’s convention, the head of the beam is on the left throughout.

Initial optimization of the linac configuration is done using the 1-d tracking code, Li-Track [9]. Here the

bunch compressors are considered point like transformations, where the U5666 can be specified.

Optimization in LiTrack is done to maximize the peak current while minimizing the ratio of the peak

current to core current, giving a solution with a flat current profile. As transverse effects are not considered

in LiTrack, the desired transformation of the current profile can be found considering adjustment of the

U5666 at BC2 only. When migrating to Elegant this total U5666 is split between BC1 and BC2, adjusting the

ratio to minimize emittance growth. We note here that Elegant simulations are done considering BC1 and

BC2 to have the same bend direction, calling for a p betatron phase advance for simplicity.

Table 2 gives Elegant simulation parameters for 4 configurations, cases I-III with 100pC and IV with 50

pC. Here the octupole ratios are chosen to minimize the projected emittance over the full beam. These

ratios could be chosen to minimize the projected emittance only over the core of the beam current.

Table 1: Initial beam parameters flattop

Parameters Value

Energy (MeV) 100 100

Laser heater ΔE (keV) 5 15

Electron beam charge (pC) 100 50

Peak current (A) 13.2 9.4

Proj. emittance (mm-mrad) 0.3 0.17

Figure 2: electron beam longitudinal phase at entrance of L1, showing current profile (blue) and energy distribution (yellow) for 100 pC (left) and 50 pC (right)


July 28, 2020 LCLSII-TN-20-05 4

Given values for the BC1 and BC2 R56 and U5666 are found from simplified analytical expressions. The

corresponding longitudinal phase spaces at the end of BC2 and hard x-ray undulator entrance are shown in

Figure 3.

Although emittance growth is well compensated after BC2, the proposed scheme is still limited by particle

losses in the collimator immediately downstream of BC1 due to strict radiation requirements. Case I gives a

configuration where losses are kept below 1% at the sacrifice of both horn suppression and emittance

compensation. For case II and III we assume that increased upstream energy collimation or running at lower

repetition rate could reduce losses to an acceptable level, focusing on achieving a flat, high peak current

profile ideal for self-seeding. Case IV shows that with the decreased initial bunch length for the 50 pC case

a configuration with no losses can be found. In all cases, remnant projected emittance growth can be

attributed to second order transverse focusing effects in the second linac section. However, the slice

emittance in the core of the beam is unaffected. This is discussed further in section 5. Case II is used as an

example case throughout the remainder of the text.

Table 2: Parameters for Elegant simulations with incoming beam from flattop temporal profile cathode laser

Parameter Case I Case II Case III Case IV

L1 (voltage per cavity [MV]) 13.3879 13.5329 14.8525 15.625

L1 RF phase [degrees] -22 -24.9 -24.9 -26.394

L1 harmonic cavity (volt per cavity [MV]) 3.041 2.97771 3.23217 4.4969

L1 harmonic cavity RF phase [degrees] -165.448 -167.074 -168.027 -163.117

Energy at BC1 [GeV] 0.25 0.25 0.265 0.260


L2 RF phase [degrees] -29.3966 -35.5773 -38.4953 -32.0556

Energy at BC2 [GeV] 1.6 1.5 1.5 1.5


L3 RF phase [degrees] 0 0 0 0

Energy at undulator entrance [GeV] 4 4 4 4

BC1 R56 [mm] (BC1 compression, bend angle [mrad])

-51.25 (2.7, 99.42)

-47.447 (2.6, 95.66)

-47.594 (2.7, 95.81)

-31.13 (2.4, 77.49)

BC2 R56 [mm] (total compression, bend angle [mrad])

-53.843 (80.2, 51.25)

-44.925 (112, 46.81)

-41.034 (157, 44.74)

-44.457 (90, 46.57)

BC1 octupole K3L (U5666 [m])

-2888.84 (2.253)

-5386.57 (3.670)

-5502.6 (3.775)

-2531.2 (0.699)


-468.92 (6.279)

-457.01 (4.244)

-446.0 (3.4465)

-334.1 (3.014)

I (undulator ent. [kA]) 1 1.5 2 1

Proj. emittance (@undulator [mm-mrad]) 1.05 0.84 0.96 0.38

Losses at BC1 collimator [%] 0.95 4.06 4.58 0.0


July 28, 2020 LCLSII-TN-20-05 5

Figure 3: Longitudinal phase space from Elegant at the end of BC2 (Left) and start of the hard x-ray undulator (Right) for case I (a), case II (b), case III (c), and case IV (d). The current profile is shown in blue with the final energy distribution in yellow.

a)

b)

c)

d)


July 28, 2020 LCLSII-TN-20-05 6

3 Cathode laser with Gaussian temporal profile For the case where the electron beam is produced by a cathode laser with a Gaussian temporal profile, the

initial current profile is peaked near the head of the beam, with significantly less third order chirp. This

distribution will not generally lead to the growth of current horns after downstream compression, but the final

current shape is typically asymmetric. Since in the LCLS-II configuration the longitudinal space charge and

longitudinal wakefields in the long bypass line play an important role to reduce the remaining chirp after the

final linac section, adjustment of higher order compression is still useful in shaping the current profile and

modifying the final longitudinal phase space. We consider an initial distribution generated by Astra

simulations of the LCLS-II injector [10]. Initial beam parameters are given in Table 3. The longitudinal

phase space at the entrance of the first linac section is shown in Figure 4. To counteract the stronger micro-

bunching instability, the amplitude of the laser heater modulation is increased to 7.5 keV. Whether this

increased microbunching instability is a physical or statistical effect remains to be seen.

In order to increase and flatten the peak current while controlling the final longitudinal phase space, it is

necessary to shift the current distribution peak towards the tail of the beam. This can be accomplished by

adjusting second order compression with the harmonic linearizer, while increasing the nominal third order

compression of the bunch compressors with the aid of the octupoles.

Table 4 again gives Elegant simulation parameters for 3 configurations. The corresponding longitudinal

phase spaces at the end of BC2 and hard x-ray undulator entrance are shown in Figure 5. Case A gives a

configuration where the current profile is ramped to produce a final longitudinal phase space with narrow

energy spread. Case B gives a configuration with a flattened current profile at the exit of BC2 with a peak

current of 2 kA. Case C gives a configuration with high peak current, but with a significant chirp at the

undulator entrance. This could be potentially utilized with a chirped taper scheme in the FEL lasing process.

None of these cases require a significant octupole kick and projected emittance growth after BC1 is small

compared with the flat beam case. Projected emittance growth from CSR is reduced by adjusting the BC2

horizontal dispersion with quadrupoles inside BC2 [11]. None of the three cases exhibit particle losses due

to the shorter incoming electron beam. Table 3: Initial beam parameters

Parameters Value

Energy (MeV) 92.36

Laser heater ΔE (keV) 7.5

Electron beam charge (pC) 100

Peak current (A) 10.76

Projected emittance (mm-mrad) 0.5 Figure 4: electron beam longitudinal phase at entrance of L1, showing current profile (blue) and energy distribution (yellow)


July 28, 2020 LCLSII-TN-20-05 7

Table 4: Parameters for Elegant simulations with incoming beam from Gaussian temporal profile cathode laser

Parameter Case A Case B Case C

L1 (voltage per cavity [MV]) 15.4023 13.7818 16.0166

L1 RF phase [degrees] -24.2386 -24.9 -32.0846

L1 harmonic cavity (voltage per cavity [MV]) 3.75 2.6701 3.71931

L1 harmonic cavity RF phase [degrees] -172.246 -179.473 -181.318

Energy at BC1 [GeV] 0.258 0.25 0.265


L2 RF phase [degrees] -33.9127 -30.2109 -35.2267

Energy at BC2 [GeV] 1.532 1.5 1.5


L3 RF phase [degrees] 0 0 0

Energy at undulator entrance [GeV] 4.03 3.99 4

BC1 R56 [mm] (BC1 compression factor, bend angle [rad])

-52.45 (3.3, 0.10058)

-70.695 (3.66, 0.11677)

-48.525 (3.33, 0.09674)

BC2 R56 [mm] (total compression factor, bend angle [rad])

-43.415 (152, 0.04602)

-50.554 (171, 0.04966)

-41.266 (286, 0.04487)


365.09 (-0.417)

900 (-1.538)

335.24 (-0.342)


73.02 (-0.734)

212.8 (-2.657)

70.14 (-0.644)

I (undulator ent. [kA]) 1.6 2.0 3.5

Proj. emittance (undulator ent. [mm-mrad]) 0.65 0.74 0.87

Losses at BC1 collimator [%] 0.0 0.0 0.0


July 28, 2020 LCLSII-TN-20-05 8

Figure 5: Longitudinal phase space from Elegant at the end of BC2 (Left) and start of the hard x-ray undulator (Right) for case A (top), case B (middle) and case C (bottom). The current profile is shown in blue with the final energy distribution in yellow.


July 28, 2020 LCLSII-TN-20-05 9

4 Longitudinal shaping An electron passing through an octupole magnet with negligible vertical offset relative to the magnetic center, will receive a horizontal kick depending on its horizontal offset given by:

𝑥!~− "!!!

#"$𝐿%𝑥& ≡ − '

#𝐾&𝐿%𝑥& (1)

Here L0 is the octupole length and K3 is the octupole’s geometric strength. Placing an octupole at a point of

high transverse dispersion, we assume an electron’s transverse offset at the octupole entrance is dominated

by its energy offset from the central energy. Considering placing the octupole in the center of a chicane, the

transverse offset at the octupole entrance is then given by the R16 from a simple dogleg, leading to an energy

dependent kick:

𝑥()* = 𝑅'#𝛿~ − 𝜃(𝑙+ + 𝑙,)𝛿 → 𝑥()*! = '#𝐾&𝐿%𝜃&(𝑙+ + 𝑙,)&𝛿& (2)

Here lb is the dipole length, ld is the drift length and q is the bend angle associated with the chicane. After

transport through the remainder of the chicane, the resultant path length difference from the octupole kick is

given by the R52 from a simple dogleg.

𝑧-()* = 𝑅./𝑥()*! = − '#𝐾&𝐿%𝜃0(𝑙+ + 𝑙,)0𝛿& (3)

Considering the nominal approximate values for 1st, 2nd and 3rd order longitudinal dispersion from a chicane,

including the octupole gives:

𝑅.#~− 2𝜃/ 3𝑙, +/&𝑙+4 , 𝑇.##~−

&/𝑅.#,𝑈.###~−

'#𝐾&𝐿%𝜃0(𝑙+ + 𝑙,)0 + 2𝑅.# (4)

We can gain some insight into the role of higher order compression in the growth and suppression of current

horns by considering the transformation of the initial current profile in the limit of zero initial energy spread.

For an initial current profile, I0(z0), transport through a dispersive element, z1= z0+f(z0), gives a final current:

𝐼'(𝑧')~ 312"12#43'𝐼%[𝑧%(𝑧')]~𝐶' ∗ (1 + 𝑐''𝑧%(𝑧') + 𝑐'/𝑧%(𝑧')/ +⋯)3' ∗ 𝐼%[𝑧%(𝑧')] (5)

𝐶' ≡ @12"12#A2#4%

B3'

,𝑐'' ≡ 𝐶' ∗ @1$2"12#$

A2#4%

B,𝑐'/ ≡'/𝐶' ∗ @

1%2"12#%

A2#4%

B

From the above expression we see that if the contribution from higher order compression approaches zero

within the initial current profile, current horns can form. This can be avoided by adjusting the second order

chirp and U5666 to reduce the 2nd and 3rd order compression terms, c11 and c12. Perhaps more importantly,

values for c11 and c12 can be found that give a desired current profile without giving rise to current horns.

Maintaining values for the higher order compression terms, the linac parameters can be varied and the peak

current can be scaled by the overall compression factor, C1, while approximately maintaining the current

profile, considering only the linear transformation of z:

𝑧%(𝑧')~𝐶' ∗ 𝑧' ≡ 𝜁 → 𝐼'(𝜁) = 𝐶' ∗ (1 + 𝑐''𝜁 + 𝑐'/𝜁/)3'𝐼%[𝜁] (6)


July 28, 2020 LCLSII-TN-20-05 10

As stated earlier, initial optimization of the final current profile was done in the 1-d tracking code LiTrack

and confirmed with Elegant. Given linac parameters from this optimization, the initial electron beam chirp,

and linac wakefields calculated analytically from [12], a simple model can be considered keeping terms up

to 3rd order. Provided that current horns are suppressed, CSR and longitudinal space charge can be ignored.

For case II, this analysis gives values for the higher order compression terms, c11 = 1.42 and c12 = 25051.7

after BC1, and c11= 985.6 and c12 = 624007 after BC2. If the current profile after BC1 is approximately

maintained then the linac wakefields in L2 can be scaled with the BC1 compression factor, and the parameter

space can be explored further while maintaining the final current profile. Figure 6 shows comparison of the

various simulation methods for case II as an example. The same configuration with octupoles removed is

shown for reference.

5 Emittance correction As mentioned earlier the double-octupole scheme is used to correct projected emittance growth from the

transverse kick associated with a single octupole. In order to better understand the emittance compensation

scheme we can approximate the transverse kick from the two octupoles at pertinent locations. The expression

from equation 2 can be written in terms of the beam coordinate in the limit that d is dominated by the

correlated chirp. At BC1 the linear chirp depends on the initial chirp, linac wakefields, harmonic linearizer

and RF phase. However, we can express this chirp in terms of the BC1 compression factor, C1, and the

chicane bending angle, q1.

𝑥′(𝑧) = !"𝐾!𝐿! )

#56$#76#56$

8976:

*%

+ !&6,%+'6(!'6$!

,%𝑧% (7)

Here K1 refers to the geometric strength of the first octupole and L1 is the octupole length, lb1 is the chicane

magnet length and ld1 is the chicane drift length. It should be noted that z refers to the transformed beam

coordinate after transport through half of the BC1 chicane, Figure 7a.

Figure 6: Left: current profile from LiTrack (green), Elegant (blue), Elegant with octupoles off (red), simplified transformation to 3rd order (yellow), and current transformation from equation 5 (dashed). Right: comparison of longitudinal phase space from Elegant with octupoles on (blue) and off (red)


July 28, 2020 LCLSII-TN-20-05 11

After transport through the subsequent linac section, assuming the electron beam goes through an n*p

betatron phase advance, we can write the transformation of the first octupole kick at the entrance of the

second octupole in terms of the energy and beta function at the BC1 octupole, E1 and β1, and energy and beta

function at the BC2 octupole, E2 and β2.

𝑥)(𝑧) = −(−1)* !"𝐾!𝐿! )

#56$#76#56$

8976:

*%

+ !&6,%+'8∗('6(!)

'8$'6,%/.6/6.8/8

𝑧% (8)

Again the octupole kick is written in terms of the compressed longitudinal beam coordinate after transport

through half of the BC2 chicane in terms of the total compression factor, C2, Figure 7b.

The kick provided by the second octupole can again be written expressing the linear chirp at the BC2 entrance

in terms of the BC1 and total compression factors. If the bend direction of the BC2 chicane is opposite to

that of BC1 then the sign of this kick will flip:

𝑥)(𝑧) = − !"𝐾0𝐿0 )

#58$#78#58$

8978:

*%

+ !&8,%+'8('!'8$'6

,%𝑧% (9)

Here K2 refers to the geometric strength of the second octupole, L2 is the octupole length, lb2 is the chicane

magnet length, ld2 is the chicane drift length, and q2 is the chicane bend angle, Figure 7c. Setting the total

kick to zero, given by the sum of equations 8 and 9, gives an approximate condition on the ratio of octupole

strengths for correcting the projected emittance growth.

18281626

= )(#56$#76)∗3#58$

8978: 4

(#58$#78)∗3#56$8976: 4*%

+&8&6,%+'8∗('6(!)

'8('6,%/.6/6.8/8

(10)

Figure 7: (a) Kick after BC1 octupole. (b) BC1 octupole kick transported to BC2 octupole entrance. (c) BC2 octupole kick with BC1 octupole off. (d) Combined kick from BC1 and BC2 octupole with minimized projected emittance. (e) Combined octupole kick with analytically calculated projected emittance minimized. Analytical estimates are shown in red.

a) b)

c) d) e)


July 28, 2020 LCLSII-TN-20-05 12

A more detailed discussion of the emittance correction scheme can be found in [13]. Figure 7d shows the

corrected octupole kick after varying K2/K1 to minimize the projected emittance in Elegant simulations for

the case II configuration. Here we see that additional emittance growth in the head of the beam, possibly

from CSR, gives an optimal K2/K1 ratio differing from that given by the above analysis. Figure 7e shows

the residual kick with the K2/K1 ratio given by equation 10, noting that for this configuration, β1/β2 ~ 1/3.

Varying the K2/K1 ratio while maintaining the total U5666, we see the slice emittance after BC2 is preserved

in the core of the beam, varying only in the head of the beam. Varying the betatron phase advance between

octupoles, again the core slice emittance is preserved, however there is more significant change in the head.

This could possibly mitigated by properly optimizing the lattice for each case, Figure 8.

6 Particle losses For the high power LCLS-II facility, we have several groups of halo collimator systems to remove the dark

current. As stated earlier, in the example of the 100 pC flattop laser case, we observed particle losses at the

collimator located downstream of BC1. Figure 9 shows the location of particles in the incoming phase space

that will be collimated for the case II configuration. From this we see that a significant portion of these lost

particles could potentially be truncated earlier using an upstream energy collimator in the laser heater area

where the energy is still low. As seen in case IV of Table 2, this is also accomplished by reducing the beam

charge, in turn reducing the initial bunch length.

Losses can also be reduced by adjusting the linac configuration. Again using case II as an example, we can

vary the BC1 energy, chirp and R56 to minimize the BC1 octupole kick given by equation 7. This is done

while maintaining the total compression after BC2 and higher order compression terms given by equation 5,

to ensure that the final current profile is maintained. The emittance correction condition given by equation

10 is also maintained.

Figure 8: (Left) Slice emittance at BC2 exit varying the ratio between BC1 and BC2 octupole strengths. (Right) Slice emittance varying the betatron phase advance between BC1 and BC2 octupoles.


July 28, 2020 LCLSII-TN-20-05 13

Figure 10 shows the octupole kick, normalized to the kick associated with case II, for BC1 energy 230

MeV, 250 MeV and 270 MeV. Solutions with linac parameters outside of LCLS-II specifications are cut.

From this we see a general trend that the BC1 kick is reduced for larger chirp, smaller R56, and larger BC1

energy. Elegant simulations with the solution giving minimal kick, show reduction of particle losses to

~3%. This could possibly be reduced further at the expense of some horn suppression and emittance

correction, as is shown in case I of Table 2.

Figure 9: (Left) Longitudinal phase space at beginning of L1 linac (blue) showing particles transported through collimators (yellow). (Right) current profile at beginning of L1 (blue) showing current profile of non-collimated particles (yellow).

Figure 10: BC1 octupole kick normalized to the octupole kick associated with case II, varying the BC1 R56 and electron beam chirp at the BC1 entrance for BC1 energy 230 MeV (left), 250 MeV (center) and 270 MeV (right). For reference the point associated with case II is indicated by the black circle. Plots are offset vertically to align where the normalized kick = 1 for each energy.


July 28, 2020 LCLSII-TN-20-05 14

7 Further discussion/conclusions

Initial studies of the presented scheme considered a second smaller chicane installed immediately

downstream of BC2, Figure 11. This setup simplifies the scheme as there is no energy gain or collimation

between the two octupoles and was considered at one point for suppression of CSR effects [14]. However,

the space required for the betatron phase advance and additional chicane is not realizable in the current LCLS-

II configuration.

Reference:

[1] P. Emma, Reports No. SLAC-TN-05-004, No. LCLS-TN01-1, (2001)

[2] http://accelconf.web.cern.ch/fel2019/papers/thp035.pdf

[3] https://journals.aps.org/prab/abstract/10.1103/PhysRevAccelBeams.19.104402

[4] https://journals.aps.org/prab/abstract/10.1103/PhysRevAccelBeams.20.030705

[5] M. Borland, “Elegant: A Flexible SDDS-Compliant Code for Accelerator Simulation,” Advanced. Photon

Source LS-287, (2000)

[6] G. Marcus and J. Qiang, Reports No. LCLS-II-TN-17-04, (2017)

[7] J. Qiang, R. D. Ryne, S. Habib, V. Decyk, An object-oriented parallel particle-in-cell code for

beamdynamics simulation in linear accelerators, J. Comput. Phys. 163, (2000) p. 434.

[8] J. Qiang, S. Lidia, R. D. Ryne, and C. Limborg-Deprey, A three-dimensional quasi-static model for high

brightnees beam dynamics simulation, Phys. Rev. ST Accel. Beams 9, (2006) 044204.

[9] K. Bane and P. Emma, in Proceedings of the 21st Particle Accelerator Conference, PAC2005, Knoxville,

TN, 2005. (IEEE, Piscataway, NJ, 2005), p. 4266

[10] Astra, https://www.desy.de/~mpyflo/

[11] N. Neveu et al, Reports No. LCLS-II-20-03, (2020)

[12] https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.120.014801

Figure 11: Double chicane configuration with pi phase advance (not shown)


July 28, 2020 LCLSII-TN-20-05 15

[13] Y. Nosochkov, Reports No. LCLS-II-20-04, (2020)

[14] http://inspirehep.net/record/469134/files/30014213.pdf?version=1 (chapter 7.4.1)

lcls-ii-tn-20-05 · 2020. 7. 28. · lcls-ii technical note july 28, 2020 lclsii-tn-20-05 4 given...

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